Question

A toroid is a long coil of wire wound over a circular core. The major radius and cross-sectional radius of the toroid are R and r, respectively. The coefficient of mutual induction of the toroid is ______.

(The magnetic field in it is uniform, N = number of turns, R >> r, μ₀ = permeability of free space)

Options

• `(mu_0NR)/(2r)`

• `(mu_0N^2R^2)/(2r)`

• `(mu_0Nr)/(2R)`

• `(mu_0N^2r^2)/(2R)`

Answer 1

A toroid is a long coil of wire wound over a circular core. The major radius and cross-sectional radius of the toroid are R and r, respectively. The coefficient of mutual induction of the toroid is `unerlinebb((mu_0N^2R^2)/(2r))`.

Explanation:

The coefficient of mutual induction is given by,

`M = (mu_0N_1N_2A)/l` ...........(i)

where µ₀ is the permeability of free space,

N₁ is the number of turns in the primary coil,

N₂ is the number of turns in the secondary coil,

A is the common area of cross-section and l is the length of coils.

Thus, for toroid, the Eq. (i) is given as,

M = `(mu_0N . N . piR^2)/(2pir) = (mu_0N^2R^2)/(2r)`

`[(∵ N_1 = N_2 = N),(A = piR^2","  l = 2pir)]`

where R is the major radius and r is the minor radius.